## Abstract

Let p ≡ ± 1 (mod 8) be a prime which is a quadratic residue modulo 7. Then p = M^{2} + 7N^{2}, and knowing M and N makes it possible to "predict" whether p = A^{2} + 14B^{2} is solvable or p = 7C^{2} + 2D^{2} is solvable. More generally, let q and r be distinct primes, and let an integral solution of H^{2}p = M^{2} + qN^{2} be known. Under appropriate assumptions, this information can be used to restrict the possible values of K for which K^{2}q = A^{2} + qrB^{2} is solvable and the possible values of K′ for which K′^{2}p = qC^{2} + rD^{2} is solvable. These restrictions exclude some of the binary quadratic forms in the principal genus of discriminant -4qr from representing p.

Original language | English |
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Pages (from-to) | 263-282 |

Number of pages | 20 |

Journal | Journal of Number Theory |

Volume | 19 |

Issue number | 2 |

DOIs | |

State | Published - Oct 1984 |

### Bibliographical note

Funding Information:* This research received some support from National

### Funding

* This research received some support from National

Funders | Funder number |
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Russian Science Foundation | GP8973 |